Usual Finite Element Research Articles

A FEM–DtN formulation for a non‐linear exterior problem in incompressible elasticity

AbstractIn this paper, we combine the usual finite element method with a Dirichlet‐to‐Neumann (DtN) mapping, derived in terms of an infinite Fourier series, to study the solvability and Galerkin approximations of an exterior transmission problem arising in non‐linear incompressible 2d‐elasticity. We show that the variational formulation can be written in a Stokes‐type mixed form with a linear constraint and a non‐linear main operator. Then, we provide the uniqueness of solution for the continuous and discrete formulations, and derive a Cea‐type estimate for the associated error. In particular, our error analysis considers the practical case in which the DtN mapping is approximated by the corresponding finite Fourier series. Finally, a reliable a posteriori error estimate, well suited for adaptive computations, is also given. Copyright © 2003 John Wiley & Sons, Ltd.

An Efficient Adaptive Procedure for Three-Dimensional Fragmentation Simulations

We present a simple set of data structures, and a collection of methods for constructing and updating the structures, designed to support the use of cohesive elements in simulations of fracture and fragmentation. Initially, all interior faces in the triangulation are perfectly coherent, i.e. conforming in the usual finite element sense. Cohesive elements are inserted adaptively at interior faces when the effective traction acting on those faces reaches the cohesive strength of the material. The insertion of cohesive elements changes the geometry of the boundary and, frequently, the topology of the model as well. The data structures and methods presented here are straightforward to implement, and enable the efficient tracking of complex fracture and fragmentation processes. The efficiency and versatility of the approach is demonstrated with the aid of two examples of application to dynamic fracture.

Direct and indirect approximations to positive solution for a nonlinear reaction-diffusion problem. I. Direct (variational)

We consider a nonlinear, second-order, two-point boundary value problem that models some reaction-diffusion precesses. When the reaction term has a particular form, \(f(u)=u^{3}\), the problem has a unique positive solution that satisfies a conserved integral condition. We study the bifurcation of this solution with respect to the length of the interval and it turns out that solution bifurcates from infinity. In the first part, we obtain the numerical approximation to the positive solution by direct variational methods, while in the second part we consider indirect numerical methods. In order to obtain directly accurate numerical approximations to this positive solution, we characterize it by a variational problem involving a conditional extremum. Then we carry out some numerical experiments by usual finite elements method.

Splitting modulus finite element method for orthogonal anisotropic plate benging

Splitting modulus variational principle in linear theory of solid mechanics was introduced, the principle for thin plate was derived, and splitting modulus finite element method of thin plate was established too. The distinctive feature of the splitting model is that its functional contains one or more arbitrary additional parameters, called splitting factors, so stiffness of the model can be adjusted by properly selecting the splitting factors. Examples show that splitting modulus method has high precision and the ability to conquer some ill-conditioned problems in usual finite elements. The cause why the new method could transform the ill-conditioned problems into well-conditioned problem, is analyzed finally.

Approximating Local Averages of Fluid Velocities: The Stokes Problem

As a first step to developing mathematical support for finite element approximation to the large eddies in fluid motion we consider herein the Stokes problem. We show that the local average of the usual approximate flow field u h over radius δ provides a very accurate approximation to the flow structures of O(δ) or greater. The extra accuracy appears for quadratic or higher velocity elements and degrades to the usual finite element accuracy as the averaging radius δ→h (the local meshwidth). We give both a priori and a posteriori error estimates incorporating this effect.

Modeling flow and heat transfer in tubes using a fast CFD formulation

An approach to study turbulent flow and conjugate heat transfer in tubes is proposed in this work. Instead of using the usual finite element or finite volume methods, this formulation applies a different technique that calculates both for the flow and heat transfer. It discretizes the flow in the radial direction using a fourth order finite differences method, which is more accurate than the traditional second order schemes. Using this technique, a system composed of several ordinary differential equations for the temperature and a set of linear equations for the velocities and pressure gradient is obtained. The equations are then integrated in the axial direction using a fourth order Runge–Kutta method. The values of viscosity, density and thermal conductivity are dependent on temperature, which makes the model suitable for the calculation of high temperature gradients, as in the case of refinery fired heaters. The turbulence is taken into account using a zero order turbulence model. A very interesting feature of this formulation is that for the laminar case the method is non-iterative, which makes it more desirable and faster than conventional computational fluid dynamics (CFD) packages.

On the Numerical Computation for Solving the Two-Dimensional Parabolic Equations by Space-Time Finite Element Method.

A space-time finite element method was presented for solving time-dependent and two-dimensional heat conduction problems. Linear hexadedron elements in a space-time domain were used as the finite elements. In this paper we present a discretization technique in which finite element approximations are used in time and space simultaneously for a relatively large time period called a time slab. The weighted residual process is used to formulate a finite element method for a space-time domain based upon the continuous Galerkin method. Numerical illustrations have indicated that the unstructured space-time algorithm provides more rapid convergence to the analytical solutions than the usual finite element analysis with discretization in space only.

815 特異要素の開発とき裂を有する平板の面内固有振動解析への応用(オーガナイズドセッション : 計算力学と構造設計)

A singular finite element (three-noded triangular and four-noded isoparametric element), which has a stress singularity, proposed by Akin was investigated in this paper. The singular elment is incorporated into a usual finite element program which can analyze eigenvalues of the two-dimensional elastic body. Using the singular element, in-plane vibration problems of the plate with an edge crack were analyzed. From the results of the numerical simulation, it was confirmed that the singular element was useful for the dynamic problem.

P-adaptive finite element analysis of unbounded electromagnetic wave problems

The boundary that artificially truncates an unbounded domain in the finite element analysis of an electromagnetic wave problem necessarily introduces a numerical error. The adaptive technique proposed reduces this boundary error, as well as the usual finite element discretisation error. The scattering or radiating object and its immediate surroundings are meshed with hierarchal finite elements. Outside, a thick layer of free space is meshed with relatively few hierarchal wave-envelope elements. During p-adaption, increasing the order of the wave-envelope elements reduces the boundary error. Moreover, this reduction in boundary error is selective: in directions of strong radiation, the error reduction is greater.

Orthogonality of Modes of Structures When Using the Exact Transcendental Stiffness Matrix Method

This paper presents theory, physical insight and results for mode orthogonality of piecewise continuous structures, including both coincident and non-coincident natural frequencies. The structures are ones for which exact member equations have been obtained by solving the governing differential equations, e.g. as can be done for members of plane frames or prismatic plate assemblies. Such member equations are transcendental functions of the distributed member mass and the frequency. They are used to obtain a transcendental overall stiffness matrix for the structure, from which the natural frequencies are extracted by using the Wittrick-Williams algorithm, prior to using any existing method to find the modes which are examined from the orthogonality viewpoint in this paper. The natural frequencies and modes found are the exact values for the structure in the sense that the usual finite element method approximations are avoided.

Computation on transient nonlinear electromagnetic field by an improved space-time finite element method

The paper presents a suitable functional in the form of a convolution of the magnetic vector potential in the space-time domain for transient nonlinear eddy problem. And then, appropriate element equations have been derived following the usual finite element variational procedure. Furthermore, the element equations are improved. Finally, the improved space-time finite element method (STFEM) has been used to compute the transient nonlinear electromagnetic field and performances of a permanent magnet synchronous motor (PMSM). Numerical illustrations have indicated that the improved space-time finite element method is a valid method to simulate performances of PMSM.

Hybrid Optimization Strategy Beyond Local Optima in Aerospace Panel Designs

Convergence to local optima rather than the global optimum is a common shortcoming of directed seareh (gradient-based) optimization techniques. Successive convergence on better local optima is an efficient way of finding the global optimum and is achieved by the proposed hybrid global optimization strategy, which combines ideas from the improving hit-and-run and the tunneling global optimization methods. The searching technique of improving hit-and-run is adapted to find suitable starting points to converge on other (better) local optima. The suitability of candidate starting points is confirmed by the concept of the tunneling method. The strategy presented has unique features due to the underlying structural theory being exact, in the sense of differential equations being solved for the member (or element) stiffnesses so that they are transcendental functions of the load factor. Hence the buckling eigenproblem for the entire structure is a transcendental one, instead of the linear one associated with the more usual finite element method formulations. Constrained minimum-mass aerospace panel stability design problems are used to demonstrate the efficiency and robustness of the hybrid global optimization strategy.

FINITE ELEMENTS FOR MODELLING BEAMS AFFECTED BY A DISTRIBUTED LOAD

Usually a finite element with cubic deflection approximation function is applied when evaluating the stress and strain field of bar structures. But such an element only approximately evaluates the actual strain field of the bar affected by a distributed load. The improved finite elements (Fig 1, 2) with fourth and fifth-order deflection approximation functions (1), (6) and (13) are presented in the actual manuscript. The fifth-order deflection approximation function is used for modelling the beams affected by a linearly distributed load (11). The plain bending of the finite element is modelled by 5 and 6 freedom degrees. The additional 5th and 6th freedom degrees are the deflection and deviation of the middle node of element (Fig 2). The element stiffness matrices (Table 1, 2) and node force vectors are presented. The created finite elements exactly modells the stress and strain field of bars, which are affected by distributed load, and also allow to compute directly the middle section displacements of bars. It creates conditions for diminishing the volume of problems and obtaining information, which is necessary to be analysed later. The reduced finite elements (Fig 4) are created by the elimination of the internal freedom degrees. Their number of freedom degrees is decreased up to the number of freedom degrees of a usually applied finite element. But the reduced finite elements have all afore-mentioned qualities. Formulas (20) and (21) are derived expressing the middle node displacements by the final node displacements. These formulas allow to compute the middle section displacements of the bar already after the solution of equation system. The proposed reduced elements can be introduced and applied in engineering practice very easily, because their stiffness matrix coincide with the stiffness matrix of a usual bar finite element. The created elements with internal freedom degrees are very important for the problems of structures optimization with displacement constraints, because the constraint of bar middle section displacement can form just in case, when this displacement is one of the problem's unknown. Also it is very important to decrease the number of unknowns of optimization problem.

Higher order finite element solution of the one‐dimensional Schrödinger equation

The one-dimensional Schrödinger equation has been examined by means of the confined system defined on a finite interval. The eigenvalues of the resulting bounded problem subject to the Dirichlet boundary conditions are calculated accurately to 20 significant figures using higher order shape functions in the usual isoparametric finite element method. Numerical results are given for an arbitrary polynomial potential of degree M. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 71: 147–152, 1999

Equivalent loads and interpolated solution in displacements on Bernouilli-Euler beam

A method for the calculus of Bernoulli-Euler's beam is presented. This method allows the results obtained by usual finite element methods to be improve. The main advantage of the proposed method is that it can approximate the displacements, bending moments and shear in the elements, including in large elements, accurately.

Application of Free Mesh Method to Fluid Analysis. 2nd Report. Large Scale Analysis by Parallel Computation.

In this paper, Free Mesh Method, which is a kind of meshless method based on the usual finite element method, is applied to unsteady incompressible viscous flow analysis using a parallel computer. The method is well compatible with the parallel environment because the global matrices of the finite element method, mass, advection, diffusion and gradient matrices, are independently computed node-by-node, and furthermore because the node-element connectivity information is unnecessary. To solve the governing equations of fluid dynamics, the Navier-Stokes and the the continuity equations, the velocity correction method with interpolation functions for velocity and pressure having equal order was used for time integration. A message passing library was used for the parallelization and the parallel efficiency over 83% was obtained for a large scale problem with 480 000 degrees of freedom using 16 processors.

Analysis of transient nonlinear eddy current fields by space-time finite element method

A suitable functional in the form of a convolution of the magnetic vector potential in the space-time domain for transient nonlinear eddy current problem has been derived. Appropriate element equations have thus been derived following the usual finite element variational procedure. The space-time finite element method has then been used to compute the transient nonlinear electromagnetic field in a practical transformer. Numerical illustrations have indicated that the computed results are good agreement with the tested ones.

Application of the Free Mesh Method to Three-Dimensional Complex Shape.

The Free Mesh Method (FMM) is simple, accurate, and suitable for parallel processing. In this method, the connectivity between nodes and elements, that is essential in usual Finite Element Analysis, is not required. This paper describes a more efficient processing with local Voronoi diagram for preprocessing at the stage of creating a set of temporary local elements in the FMM. Adopting the surface patch model to describe analysis model, the present FMM was applied to three-dimensional complex shape. In the analysis of three-dimensional steady thermal conduction problems of complex geometry, the present method was shown to work in reasonable calculation time.

On boundary conditions in the element-free Galerkin method

Accurate imposition of essential boundary conditions in the Element Free Galerkin (EFG) method often presents difficulties because the Moving Least Squares (MLS) interpolants, used in this method, lack the delta function property of the usual finite element or boundary element method shape functions. A simple and logical strategy, for alleviating the above problem, is proposed in this paper. A discrete norm is typically minimized in the EFG method in order to obtain certain variable coefficients. The strategy proposed in this work involves a new definition of this discrete norm. This new strategy works very well in all the numerical examples, for 2-D potential problems, that are presented here. In addition to the discussion of boundary conditions, some recommendations are also made in this paper regarding strategies for refinements in order to improve the accuracy of numerical solutions from the EFG method.

Soil-Water Coupled Behavior of Heavily Overconsolidated Clay Near/At Critical State

The time dependent elasto-plastic response of a heavily overconsolidated soil has been illustrated with simple yet typical examples that a heavily overconsolidated clay under loading exhibits either hardening or softening due to the migration of pore water. In the examples studied triaxial tests are numerically simulated, where the test is considered not as a test of a soil element but as a test of the soil mass with well defined boundary conditions. In order to describe the behavior of heavily over consolidated soils, the subloading surface concept first developed by Hashiguchi and Ueno (1977) is newly applied to the original Cam-clay model. From this study, the following conclusions are obtained: (1) The subloading surface Cam-clay model describes the typical shear behavior of heavily overconsolidated soils such as the hardening procedure that occurs above the critical state line with a considerable volume expansion. (2) The hardening and softening of heavily overconsolidated soils due to the migration of pore water have non-local characteristics because of Darcy’s law. Based on this gradient nature with positive permeability, plastic instability can be well simulated by the usual finite element computation. (3) As the results of coupling with Darcy’s law, the total shear behavior of the triaxial soil specimens exhibits an apparent “time dependent” nature particularly for the “stress-dilatancy” characteristic. This can be observed equally under both undrained and drained conditions for triaxial experiments.